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harmonic oscillator∗ perucho† 2013-03-21 23:28:37 1 Introduction Harmonic oscillator is the simplest model but one of the most important vibrating system. It is often encountered in engineering systems and commonly produced by the unbalance in rotating machinery, isolation, earthquakes, bridges, building, control and atomatization devices, just for naming a few examples. Although pure harmonic oscillation is less likely to occur than general periodic oscillations by the action of arbitrary types of excitation, understanding the behavior of a system undergoing harmonic oscillation is essential in order to comprehend how the system will respond to more general types of excitation. Such harmonic oscillations may be in the form of free vibrations or by the action of an exciter force applied at some point of the system, as we shall see later. Indeed, harmonic oscillator is a single degree of freedom system (it is like a vibration module) which is tipically constituted by an ideal spring in parallel with an ideal (but not conservative) viscous damper, both attached to a mass in one end and the other one grounded. In this entry we will make an elementary but nonconventional discussion about the fundamental concepts involved in a single vibrating system. Unlike the conventional method, which is usually discussed in inherent Literature about this topic, where, by one side, the solution of the corresponding differential equation of motion is expressed in terms of circular trigonometric functions, having to deal with complex arbitrary constants (obviously depending on the initial conditions) and, on the other hand, studying separately the three cases of damping. Alternatively we shall use hyperbolic functions and their properties to obtain the solution of the mentioned differential equation. This way we gain a little in brevity and hilarity in the exposition, as we may study all the three cases of damping simultaneously, avoiding at the same time calculating complex constants. ∗ hHarmonicOscillatori created: h2013-03-21i by: hperuchoi version: h39745i Privacy setting: h1i hApplicationi h34C05i h34A30i h34-01i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 Besides preliminars, we will separate the discussion in three paragraphs namely: free damped vibrations, forced damped vibrations and mechanical resonance. 2 Preliminars The purpose of this section is concept definifitions. Response: x = x(t), the general solution of the linear differential equation involved in the motion of harmonic oscillator. We will assume x > 0 downward, like the sense of gravitatory field. Ideal spring: an unidimensional model, weightless, linear, elastic and that obeys the Hooke’s law Fs = kx, where k is an elastic constant so-called stiffness, Fs is the restitutive spring force and x an arbitrary spring elongation. Ideal damper: an unidimensional model, weightless, where the damping force is proportional to the velocity of oscillator, i.e. Fd = c ẋ, being c the damping constant. p Natural angular frequency: ωn = k/m (rad/sec), a specific property of the system; m is the mass of √ oscillator. Damping factor: ζ = c/2 √km ≡ c/2mωn (physical dimensionless). Critical damping: cc = 2 km ≡ 2mωn . Thus ζ = c/cc , and it is also called damping ratio. Static equilibrium configuration: a static position at t = 0− , reached because the action of gravitatory field over the mass of oscillator, i.e. the weight mg (g is the gravity acceleration), thus deflecting the spring a quantity δ, from its naturalPlength, so-called spring static deflection. As a result, from Newton’s equation Fx = 0, the spring force kδ (damper does not work!) due to its static deflection is balanced, at every instant, by the weight of oscillator and therefore both forces do not appear in the dynamical equation. 3 Free Damped Vibrations With the help P of an oscillator’s free body diagram and by applying the Newton’s second law Fx = mẍ, we obtain the differential equation of motion mẍ + cẋ + kx = F (t), (1) being F (t) the so-called exciter force. It is convenient to introduce ωn and ζ. So that (1) becomes ẍ + 2ζωn ẋ + ωn2 x = f (t), (2) where f (t) = F (t)/m. But in this section we are interested in free vibrations. So that we set f (t) ≡ 0 and thus we get the associate homogeneous differential equation ẍ + 2ζωn ẋ + ωn2 x = 0. (3) In this case, about free oscillations, x means an arbitrary displacement from the static equilibrium position. Let us suppose initial conditions x(0) = x0 , ẋ(0) = 2 ẋ0 . These conditions may be interpreted as follows: from the static equilibrium position we give an initial displacement x0 and an initial velocity ẋ0 to the mass m and then we let it free at instant t = 0. Accordingly, the characteristic equation of (3) is s2 + 2ζωn s + ωn2 = 0, (4) whose roots are given by s1 = −ζωn + ωn p ζ 2 − 1 ≡ −ζωn + ωd , (5) p (6) s2 = −ζωn − ωn ζ 2 − 1 ≡ −ζωn − ωd , p where we are defining ωd := ωn ζ 2 − 1 which is called frecuency of damped oscillation. In the nullspace of the linear operator associate to (3), we find the general solution x(t) = Aes1 t + Bes2 t =⇒ ẋ(t) = As1 es1 t + Bs2 es2 t . (7) By substituting the initial conditions in (7), we obtain the symmetric linear system of equations x0 = A + B, ẋ0 = As1 + Bs2 . (8) From (5), (6) and solving (8) for A and B, leads to A= ẋ0 + (ζωn + ωd )x0 , 2ωd B=− ẋ0 + (ζωn − ωd )x0 , 2ωd (9) which substituted into the solution (7) and after an elementary algebraic manipulation, leads to eωd t + e−ωd t eωd t − e−ωd t e−ζωn t x(t) = + ωd x0 (ẋ0 + ζωn x0 ) , (10) ωd 2 2 that, in terms of hyperbolic functions, it is expressed as x(t) = e−ζωn t {(ẋ0 + ζωn x0 ) sinh ωd t + ωd x0 cosh ωd t} . ωd (11) It is now easily seen that the general solution of the homogeneous equation (3) satifies limt→∞ x(t) = 0, since we are dealing with a transitory solution which agree with the fact that an exciter force f (t) is not present. Damping factor lets a classification about the kind of damping in concordance with the values that ζ may take. Thus, overdamped case, 1. ζ > 1 =⇒ 2. ζ = 1 =⇒ critical damping case, 3. 0 < ζ < 1 =⇒ underdamped case. Let us proceed to study all the three cases. 3 1. ζ > 1. Overdamped case. The response is nonperiodic and is given directly by (11), i.e. x(t) = e−ζωn t {(ẋ0 + ζωn x0 ) sinh ωd t + ωd x0 cosh ωd t} . ωd (12) Roots s1 > s2 ∈ R. Hence es1 t , es1 t are linearly independent and form a basis in the above mentioned nullspace. 2. ζ = 1. Critical damping case. The response is also nonperiodic. In addition, s1 = s2 < 0, so that es1 t , es2 t are linearly dependent and therefore they do not form a complete basis in the space of solutions of the differential equation (3). Nevertheless, we may find out a second solution linearly independent because in this case ωd ≡ 0, and recalling (11) we note that (by applying L’Hospital), lim ωd →0 sinh ωd t = t. ωd Thus, the general solution for critical damping may be expressed as x(t) ≡ xc (t) = e−ωn t {x0 + (ẋ0 + ωn x0 )t} . (13) 3. 0 < ζ < 1. Underdamped case. Since √ in this case ωd ∈ C, we redefine p it as ωd = iωn 1 − ζ 2 := iωd0 , i = −1. Recalling the hyperbolic equations sinh ωd t = sinh iωd0 t = i sin ωd0 t, cosh ωd t = cosh iωd0 t = cos ωd0 t which substituted into (11) gives the general solution on this case. That is, −ζωn t ẋ0 + ζωn x0 0 0 x(t) = e sin ωd t + x0 cos ωd t . (14) ωd0 This solution is clearly harmonics. Alternatively, we may introduce definitions about the amplitude X of the response and the phase angle φ. They are, s 2 ẋ0 + ζωn x0 ωd0 x0 2 X := x0 + , tan φ := · (15) ωd0 ẋ0 + ζωn x0 With these definitions placed into (14), it becomes x(t) = Xe−ζωn t sin (ωd0 t + φ). 4 (16) Forced Damped Vibrations In this case an exciter force is present. Many useful and important applications, a few mentioned above in the Introduction, agree with an excellent degree of accuracy if we assume an intensive periodic exciter force (exciter force by mass 4 unit) on the form f (t) = f0 cos ωt1 , being f0 := F0 /m the amplitude of the intensive exciter force, and ω the frecuency of the exciter force. In such case, the differential equation of motion (2) takes the form ẍ + 2ζωn ẋ + ωn2 x = f0 cos ωt. (17) It is well-known, from the linear differential equations theory, that the general solution of (17) may be expressed as the superposition of an homegeneous or transitory response and a particular or permanent response, i.e. x(t) = xh (t) + xp (t). (18) The general homogeneous solution xh already have been stablished, so that in this section we focus our attention in find out a particular solution xp . We search that solution on the form xp (t) = C cos ωt + D sin ωt. (19) Its first and second time derivatives are ẋp (t) = −Cω sin ωt + Dω cos ωt =⇒ ẍp (t) = −Cω 2 cos ωt − Dω 2 sin ωt. (20) Next we substitute (19) and (20) into (17), setting x(t) = xp (t) and by collecting terms in cos ωt and sin ωt, we have {(ωn2 − ω 2 )C + (2ζωn ω)D} cos ωt + {(ωn2 − ω 2 )D − (2ζωn ω)C} sin ωt = f0 cos ωt. We are dealing with a subspace of dimension 2, where the set of functions {cos ωt, sin ωt} ⊂ C[0, ∞) form an orthogonal basis in the spaces of solutions of (17). In other words, cos ωt and sin ωt are there linearly independent. Therefore, (ωn2 − ω 2 )D − (2ζωn ω)C ≡ 0, and (ωn2 − ω 2 )C + (2ζωn ω)D ≡ f0 . By solving for C and D, C= (ωn2 (ωn2 − ω 2 )f0 , − ω 2 )2 + 4ζ 2 ωn2 ω 2 D= 2ζωn ωf0 · (ωn2 − ω 2 )2 + 4ζ 2 ωn2 ω 2 These one are introduced into (19) and we obtain a particular solution of (17), i.e. the permanent response of harmonic oscillator’s forced vibrations. That is, xp (t) = (ωn2 (ωn2 − ω 2 )f0 2ζωn ωf0 cos ωt + 2 sin ωt. (ωn − ω 2 )2 + 4ζ 2 ωn2 ω 2 − ω 2 )2 + 4ζ 2 ωn2 ω 2 (21) One also may define the frecuency ratio r = ω/ωn and continue to discuss an important set of physical considerations, but that is not the sight of this entry2 . 1 Naturally, an exciter force on the form F (t) = F sin ωt is also valid, but the conclusions 0 are the same. 2 Indeed the author mostly wanted to indicate the alternative elementary method corresponding to free vibrations but, for completeness reasons, the latter two sections have been added. 5 5 Mechanical Resonance In absence of damping, i.e. ζ = 0, (21) reduces to xp (t) = f0 cos ωt. ωn2 − ω 2 (22) One realize (theoretically) that as the frequency of exciter force coincides with the natural frequency of the system, i.e. ω = ωn (or equivalently r = 1), then xp = ∞. This is what is defined as mechanical resonance phenomena. In the practice, however, low values of ζ (ζ 1) make feasible high values of the permanent response and thus, this fact, is also called resonance. Such phenomena is undesirable in any system, unlike the magnetic resonance that provides beneficial applications in several fields of science. 6 References P. Fernández, Un método elemental alternativo para el estudio de las vibraciones libres amortiguadas en un oscilador armónico, (Comunicacin interna), Departamento de Mecánica, Facultad de Ingenierı́a, Universidad Central de Venezuela (UCV), Caracas, 1997. 6